# Operator algebra in the space of images

## TL;DR

This paper develops a mathematical framework using operator algebra and rigged Hilbert spaces to describe and transform images on the disk, leveraging Zernike functions and Lie algebra representations.

## Contribution

It introduces a rigorous operator algebra approach for image transformations using the structure of rigged Hilbert spaces and Lie algebra representations.

## Key findings

- Operators belong to the universal enveloping algebra of su(1,1)⊕su(1,1)
- Images are modeled within a rigged Hilbert space framework
- Physical states are represented in the Schwartz space for measurement precision

## Abstract

A consistent description of images on the disk and of their transformations is given as elements of a vector space and of an operators algebra. The vector space of images on the disk $\mathbb{D}$ is the Hilbert space $L^2(\mathbb{D})$ that has as a basis the Zernike functions. To construct the operator algebra that transforms the images $L^2(\mathbb{D})$ must be complemented and the full rigged Hilbert space $RHS(\mathbb{D})$ considered. Only this rigged Hilbert space allows indeed to write the operators of different cardinality we need to build the ladder operators on the Zernike functions that by inspection, belong to the representation $D_{1/2}^+ \otimes D_{1/2}^+$ of the algebra $su(1,1) \oplus su(1,1)$. Consequently the transformations of images are operators contained inside the universal enveloping algebra $UEA[su(1,1) \oplus su(1,1)]$. Because of limited precision of experimental measures, physical states can be always described by vectors of the Schwartz space $\S(\mathbb{D})$, dense in the $L^2(\mathbb{D})$ space where the manipulation of images is performed.